Phase-Resolved Surface Plasmon Interferometry of Graphene

Justin A. Gerber, Samuel Berweger,* Brian T. O’Callahan, and Markus B. Raschke†

Department of Physics, Department of Chemistry, and JILA, University of Colorado, Boulder, Colorado 80309, USA(Received 7 February 2014; published 30 July 2014)

The surface plasmon polaritons (SPPs) of graphene reflect the microscopic spatial variations ofunderlying electronic structure and dynamics. Here, we excite and image the graphene SPP responsein phase and amplitude by near-field interferometry. We develop an analytic cavity model that canself-consistently describe the SPP response function for edge, grain boundary, and defect SPP reflectionand scattering. The derived SPP wave vector, damping, and carrier mobility agree with the results frommore complex models. Spatial variations in the Fermi level and associated variations in dopantconcentration reveal a nanoscale spatial inhomogeneity in the reduced conductivity at internal boundaries.The additional SPP phase information thus opens a new degree of freedom for spatial and spectral grapheneSPP tuning and modulation for optoelectronics applications.

DOI: 10.1103/PhysRevLett.113.055502 PACS numbers: 81.05.ue, 07.79.Fc, 73.20.Mf

The light-matter interaction of graphene is distinct fromthat of other forms of matter due to its unique electronicband structure. The high quantum yield has already enableda range of optoelectronics and photonic applications basedon single particle excitations. However, even more unusualare the collective particle excitations in the form of Diracplasmons, typically in the infrared spectral range, with theirproperties controllable by electric field gating, doping, ormultilayer stacking [1–5]. In the long wavelength limit, theunique properties of massless Dirac fermions lead to a verylarge reduction of surface plasmon polariton (SPP) wave-length compared to the free-space excitation wavelength:λSPP=λ0 ≃ 2αEF=ðϵℏωÞ ∼ α [6], with fine structure con-stant α and Fermi energy EF. The short SPP wavelengthgives rise to a strong spatial confinement, but themomentummismatch due to the associated large in-plane wave vectorconcomitantly requires high k-vector field components forthe SPP excitation. That coupling can be achieved throughthe near field of dipolar emitters [7,8] or nanostructures suchas the pointed apex of a scanning probe tip [9–11]. Inparticular, using a tip to excite and subsequently scatter thepolarization of the Dirac plasmons into detectable far-fieldradiation, the expected deep-subwavelength SPP standingwave fromboundary reflections atmid-IR frequencies couldbe imaged using scattering-type scanning near-field opticalmicroscopy (s-SNOM) [12–15].The plasmon wavelength, its damping, and other spatial

details directly relate to the local electronic structure, whichis determined by doping or strain [1,2,16], the number oflayers, and their stacking order [5,17,18], and is furthermodified by atomic scale discontinuities at edges, grainboundaries, and defects. With the exquisite sensitivity ofthe spatial plasmon response to these parameters, near-fieldplasmon interferometry can serve as a sensitive probe forthe electronic structure and its spatial inhomogeneities,which are difficult to access by other techniques. So far,

however, near-field imaging experiments have onlyanalyzed the amplitude of the optical response [12–15].For a complete understanding of the correlation bet-

ween the optical near-field response and the underlyingelectronic structure, a full spatial characterization of theplasmon amplitude and phase is desirable. Such comple-mentary information would allow for self-consistent testsof different Dirac plasmon theories.Here, we image the full phase and amplitude response

of graphene plasmons by near-field interferometry of SPPsreflected and scattered by external (edges) and internalboundaries (folds and grain boundaries), as well as defects,of single-layer and multilayer graphene. That completeSPP response in phase and amplitude allows us to developa simple cavity model that provides a complete, self-consistent, and intuitive description of the optical physicsof the graphene plasmons and complements the morecomplex numerical electrodynamic theories. We derivethe plasmon wave vector, damping, and carrier mobility,with values in agreement with theory. We identify spatialinhomogeneities in the local electronic structure associatedwith the grain boundaries of the exfoliated graphene.In the spectral region ℏω < 2EF, where Pauli blocking

occurs, graphene exhibits a Drude-type behavior [19] andthe SPP wave vector is given by [2]:

kSPP ¼2πh2c2ϵ0κe2EFλ20

; ð1Þ

where κ is the average dielectric function of the embeddingmedia κ ¼ κ1 þ iκ2 ¼ ðϵ1 þ ϵ2Þ=2. Figure 1(a) shows thecalculated dispersion relation of the graphene SPP usingexperimental values for the SiO2 dielectric function [20]and assuming EF ¼ 0.4 eV as an example. The large in-plane momentum necessary to overcome that wave vectormismatch is provided by the evanescent near field of the tipwith apex radius r ∝ 1=ktip [9–11]. SPPs are thus launched

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http://dx.doi.org/10.1103/PhysRevLett.113.055502http://dx.doi.org/10.1103/PhysRevLett.113.055502http://dx.doi.org/10.1103/PhysRevLett.113.055502http://dx.doi.org/10.1103/PhysRevLett.113.055502

and subsequently scattered or reflected at electronic inho-mogeneities in the form of, e.g., defects [12], edges [12,13],or other structural discontinuities [14,15], as illustrated inFig. 1(b). These reflected waves, after propagating back tothe tip, interfere with other local near-field signal contri-butions and are scattered by the tip into the far field, wherethey are detected [12,13].We excite the graphene SPP at λ ¼ 10.8 μm

(ωSPP ¼ 174 THz) using a 13CO2 laser (Access Laser),off-resonance from the strong damping of the SiO2 phonons[Fig. 1(a)], and below thegrapheneoptical phonon frequencyatωph;gr ¼ 307 THz [21–24]. In the experiment, as shown inFig. 1(c), the laser is focused onto the tip of an atomic forcemicroscope (AFM,Anasys Instruments) operating in tappingmode using an off-axis parabolic mirror (NA ¼ 0.35,P ∼ 5 mW). The tip-scattered near field ENF is homodyneamplified at the HgCdTe detector (Kolmar Technologies)with the reference field Eref of the Michelson interferometerwith beamsplitter (BS). The far-field background is sup-pressed by lock-in demodulation (Zurich Instruments) at thethird harmonic of the cantilever frequency [25]. In orderto further suppress amplification of the near field by theself-homodyne background Ebg with uncontrolled phase[26], a strong reference field Eref=Ebg ≥ 10 is used [27].By collecting raster-scanned images at two orthogonalreference phases, the full complex valued tip-scattered nearfield ~A ¼ AeiΦ can be determined with low error [28].Mechanically exfoliated graphene [29] onSiO2was obtainedcommercially (Graphene Industries).Figure 2 shows a typical image of a high-aspect ratio

graphene wedge, chosen to feature both single-layer andbilayer regions, as indicated in the topography [Fig. 2(a)],with s-SNOM amplitude A ¼ j ~Aj [Fig. 2(b)] and phaseϕ ¼ argð ~AÞ [Fig. 2(c)]. The detected tip-scattered light is asuperposition of the intrinsic sample optical response that isexpected to be largely independent of tip position and the SPPwaves whose properties are a function of the local environ-ment or geometry. The amplitude thus exhibits a standing

wave pattern from the local interference of SPP reflectionfrom both edges of the wedge, with a maximum seen at adistance of λ=4 in good agreement with previous studies[12,13]. Phase and amplitude standing waves both exhibit aperiodicity of λ=2 and differ by ∼90°. A distinct feature isthe phase maximum pinned to the edge. The bilayer region(dotted line) at the tapering end of the wedge is characterizedby a decreased amplitude, in addition to reduced λSPP, as seenby a shift of the maxima in both the amplitude and phase(dashed black lines) closer to the edge of the wedge.Figure 3 shows the corresponding behavior of SPPs at

grain boundaries and folds, with the AFM topography[Fig. 3(a)], SPP amplitude [Fig. 3(b)], and phase [Fig. 3(c)]of monolayer graphene with a high density of both kinds ofline defects. As seen in bothA andϕ, plasmon reflection andstanding wave behavior are observed and are qualitativelysimilar to that of the external boundaries. The spatial signalvariations along the boundaries and within the graphenedomains are highly reproducible, including variations inSPP wavelength and amplitude across boundaries.Previous models described the measured SPP interfer-

ence images to good agreement [12–15]. However, theyrelied on complex numerical approaches and did notaddress the phase. Here, we show that the SPP oscillationscan be described in both amplitude and phase simulta-neously using a simple phenomenological cavity model

600 nm

(a) (c)(b)

Height (nm) A (arb. units) (rad)0.15 1.15 1.100 6

FIG. 3 (color online). (a) AFM topography, (b) near-fieldamplitude, (c) and near-field phase of a region of single-layergraphene with a high concentration of grain boundaries and folds.

500

nm

(a)

Height (nm) A (arb. units) (rad)0.2 1.1 200 6

(c)

i

ii

(b)

BL

SL

FIG. 2 (color online). (a) AFM topography, (b) near-fieldamplitude A ¼ j ~Aj, and (c) near-field phase ϕ ¼ argð ~AÞ of agraphene wedge. The transition from single-layer (SL) to bilayer(BL) graphene is indicated.

12

ω

AFM

HgCdTeDetector

(c) Lock-in amp.

BSCO2

(b)

SPPRe

flecti

on, R

Scattering

SiO2

PPPPPPPPPPPPPPP

Tip

ESPP

ε1

ε2

IRin

IRscat

kSPP

(a)

EF

Ligh

tSP

Pω (

1013

Hz)

10

15

20

25

30

k (106 m-1)0 1 20 40 60 80 100

ktip

ω

ω

FIG. 1 (color online). (a) Graphene SPP dispersion relation forEF ¼ 0.4 eV. Inset: Pauli blocking arising from a doping-inducedFermi level shift. (b) Illustration of tip-induced SPP excitationand subsequent interference due to the emission of scattered andreflected SPP waves. (c) Schematic of the experimental setup.

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with no independent parameters, as shown schematically inFig. 4(a). The tip-scattered near-field response of grapheneis the sum of a nonresonant dielectric contrast contribution~ψgr, a resonant local tip-induced SPP term ~ψSPP;0, and thereflected SPP fields ~ψSPP;i, as

Ψgr ¼ ~ψgr þ ~ψSPP;0 þX

i

~ψSPP;i; ð2Þ

eachwith respective amplitudeand relative phase. In addition,we consider a SiO2 substrate near-field response ~ψ sub.We describe ~ψSPP;i ¼ ~Ri× ~ψSPP;0 expf−2ReðkSPPÞriðγþ iÞgwith decay constant γ, distance between tip and reflection ri,and complex valued scattering coefficient ~Ri.In order to account for the finite size of the tip apex

generating a spatially averaged near-field response, we use aweighting function Θ convolved with the spatially varyingoptical response Ψ to simulate the s-SNOM amplitude~AðrÞ ¼ ðΨ � ΘÞðrÞ. As discussed below, we find that treat-ing Θ as Gaussian to approximate the evanescent nature ofthe tip near field, peaked at r, with awidth of 11 nm tomodelthe tip radius, provides good agreement with the exper-imental results.Figure 4 shows the simulated spatial distribution of SPP

amplitude [Fig. 4(b)] and phase [Fig. 4(c)] for a stationarypoint source (tip) located at a fixed distance from a reflectingstraight boundary located on the left edge (green line). Forall calculations, we use ~ψgr ¼ 0, ~ψSPP;0 ¼ 1, ~R ¼ −1, andγ ¼ 0.1. The line cut [Fig. 4(d)] taken along the dashed lineseparating Figs. 4(b) and 4(c) shows the propagation of theSPP away from the tip and its subsequent perturbation by thereflected SPP. Scanning the tip then results in a parallel stan-ding wave pattern in spatial s-SNOM amplitude [Fig. 4(e)]and phase [Fig. 4(f)], as also seen experimentally. From theline cut [Fig. 4(g)], we see a standing wave period of λ=2 forboth phase and amplitude in agreement with experiment(for details, see the Supplemental Material [30]).

We first compare the measured s-SNOM signal with ourmodel by examining line cuts in Fig. 2. Figure 5 shows theexperimental phase (solid red line) and amplitude (solid blueline) along dashed lines (i) and (ii) in Fig. 2, with Figs. 5(a)and 5(b) corresponding to single-layer and bilayer graphene,respectively. Graphene edges on both sides are assumedto have identical reflection and decay parameters for ~ψSPP;i.The dashed grey lines show the results of the modelcalculations, simultaneously reproducing for just a singleparameter set for the single layer and the bilayer, respec-tively, all the main spatial features both in amplitude andphase. The only exception is a larger than predicted decreasein amplitude at the graphene edge, as discussed below.The best agreement between theory and experiment is

obtained for a reflection coefficient of ~R ≈ −1, correspondingto a π phase shift, with a SPP wavelength of λSPP ¼ ð260�10Þ nm and γ ¼ 0.25� 0.04 for the single layer [Fig. 5(a)].For the bilayer [Fig. 5(b)], we find a shorter wavelength ofλSPP ¼ ð190� 10Þ nm, as discussed below, yet the samedamping and reflection coefficients. We further find a phasedifference of ∼65° between ~ψgr þ ~ψSPP;0 and SiO2 substrateresponse ~ψ sub for both single-layer andbilayer graphene. Thisphase shift is less than the 90° expected between the resonantSPP and nonresonant substrate response. However, becausegraphenedoes not entirely screen the tip-substrate interaction,signal contributions from the underlying SiO2 reduce theoverall phase shift. While the phases for bilayer and single-layer graphene are identical, an overall smaller amplitude of~ψgr þ ~ψSPP;0 is found for bilayer graphene.To model the observed spatial SPP behavior at the

different internal interfaces, we examine the line cut inFig. 3(b) (dashed red line) as an example. The resultingphase and amplitude traces are shown in Fig. 6(a) togetherwith the result of themodel (dashed grey lines). Note that theleft (right) sides of the boundary have different SPP wave-lengths of λSPP ¼ 240 nm (260 nm), as well as differentreflection coefficients of ~R ¼ 0.45 (0.55). The differentwavelengths and thus wave vectors indicate a possibledifference in electronic structure on either side of the lineardefects in that region. Variations in reflection coefficientscan be explained by the change in SPP wave vector acrossthe boundary, where momentum conservation facilitates

0 0

1 2

-π

π

-π

π-π/2

-π/2

π/2

π/2

Arg

(Ψ)

Arg(Ψ

)

Arg(Ψ

)

(b)

(c)

(d)

(f)

(g)

R = ReiθSPP,1~ SPP,0

ψsub~ gr

(a)

(e)

|Ψ|

| Ψ|

Tip

0

0 0

1

|Ψ|

0

2

Distance (λ) Distance (λ)

λ

FIG. 4 (color online). (a) Illustration of the graphene SPP cavitymodel. Calculated SPP distribution of (b) amplitude, (c) phase,and (d) the corresponding line cut along the dashed line, withlocal tip excitation and reflection from a boundary at the leftedge. The resulting spatial standing wave SPP map of s-SNOM(e) amplitude, (f) phase, and (g) line cut when scanning the tip.

(ra

d)A

(ar

b. u

nits

)

0.20.40.60.8

0.20.6

11.4

0 200 400 600Distance (nm)

(a) (b)

200 400 600

i ii

FIG. 5 (color online). Line cuts of phase (solid red line) andamplitude (solid blue line) taken along the dashed lines in Fig. 2with (i) shown in (a) and (ii) in (b). Dashed lines are fits to thecavity model for a single parameter set.

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transmission from the high-momentum to low-momentumside similar to material interfaces in conventional optics.To determine and model possible local variations in elec-

tronic properties that underly the complex optical response,Fig. 6(b) shows the relationship between λSPP and maximumamplitude Amax extracted from both sides of a series ofrepresentative boundaries indicated by the dashed black linesinFig. 3(b).A clear correlationof an increase inSPP intensityAmax with an increase in λSPP is observed. Since Amax isdirectly related to the reflection coefficient, this correlationis due to the reduced reflection of high-momentum SPPs.Using Eq. (1), we can relate that variation in SPP

wavelength to local variations in the Fermi level. Shownin Fig. 6(c) are the calculated SPP dispersion relations ofgraphene on SiO2 for different values of EF. The bluesquares indicate the range of wavelengths from Fig. 6(b),thus corresponding to spatial variations in EF of up to0.2 eV. The Fermi level directly relates to the dopingconcentration n as EF ¼ ℏvFkF, with Fermi velocity vF ≈1 × 106 m=s and Fermi momentum kF ¼

ffiffiffiffiffiffiπn

p[31]. Our

spatial variations in Fermi level of EF ≈ 0.4–0.6 eV thuscorrespond to n, ranging from 1.2 to 2.6 × 1013 cm−2.The large variation in doping across the boundaries

indicates that the reduced conductivity of boundaries[14,32] prevents charge equilibration between adjacentsides. Unlike the charge carriers themselves, the SPPs ascollective excitations are able to traverse such potentialbarriers as seen by the nonunity reflection coefficients.We then determine the carrier mobility from the exper-

imentally obtained damping γ. By correcting for the super-position of Ohmic damping and radial decay from theexcitation point, we obtain an Ohmic SPP decay constantof γp ¼ 0.12� 0.04. Using γp ≈ σ1=σ2 þ κ2=κ1, with gra-phene conductivity σ ¼ σ1 þ iσ2 [12] and κ2=κ1 ¼ 0.04, thisresults in a carrier relaxation rateΓ¼σ1ω=σ2¼ð14�4ÞTHz.The relaxation rate then relates to themobilityμ ¼ eνF=ΓℏkF[2], giving μ ¼ 1.6 × 103 cm2V−1 s−1.Our derived mobility is in good agreement with previous

s-SNOM measurements [12,13] but lower than typicalvalues for exfoliated samples, as determined by transportmeasurements [33]. Similarly, our values forEF and n agree

with values derived by s-SNOM [12,13], but they and thevariations seen due to charge pooling are larger thanexpected [31,33–35]. Recent work has attributed the dis-crepancy in damping rates and thusmobility to the large SPPfrequency and associated wave vectors, which result inincreased in impurity scattering and therefore deviationsfrom assumed dc values [36]. The surprisingly large Fermienergy has not been addressed in this work.Our results provide insight into the spatial phase behavior

of graphene SPPs. The phase is robust to changes in signalintensityduetothepresenceof,e.g.,bilayergraphene,asseeninFig. 2, or the presence of surface contaminants that otherwisereduce the amplitude signal (see Supplemental Material [30]).The optical phase also provides an additional constraint forour cavity model. In particular, the absence of independentparameters used to describe the amplitude and phase under-scores its validity.Despite its simplicity, good semiquantitativeagreement is found between the model and the data includingthe phase relationship between the amplitude and phase.However, the overall agreement in the amplitude near theedgesisseentodecreasewithdipsinthemodelandspatialshiftsin the oscillation extrema. This can be attributed to changesin local electronic structure [12] and field variations near theedges [13] and emphasizes the high sensitivity of SPP near-field interferometry to such small inhomogeneities.We note that our parameters do not reproduce the dual

peaks resulting from the localized mode indicated nearthe wedge terminus by a white arrow in Fig. 2(b). It isexpected that gradually tapered wedges as studied hereexhibit phenomena similar to nanoribbons, where a width-dependent reflection phase has been predicted [37].However, we find a constant reflection coefficient [24],as evidenced by the interference maximum in the phase atthe graphene edge which we see to abruptly transform intoa localized mode which can only be reproduced through theuse of ~R ¼ 1, as noted previously [13].In addition to the long range Dirac plasmons readily

observed in the Pauli-blocking regime, additional acousticand nonlinearmodes exist [38–40].While thesemay be diffi-cult to observewith optical means due to the strong dampingand large momenta required, these may be observable using

0 200-200Distance (nm)

φ (r

ad)

A (arb. units)0.3

0.5

0.4

0.5

0.6(a)

(b)

0.6 0.8 1Amax (arb. units)

200

260

320

λ sp (n

m)

λsp/2

Amax

(c)

100

200

400

300

10.5 1110

λ0 (µm)

λsp (nm

)

0 200-200

0.6 eV

0.5 eV

0.4 eV

EF= 0.3 eV

λ0

FIG. 6 (color online). (a) Line cut along thedashed red line in Fig. 3 showing phase (solidred line) and amplitude (solid blue line) alongwith modeled response (dashed grey lines).(b) Plot of SPP wavelength vs amplitudeextracted from both sides of the linear defectsfrom Fig. 3. The dashed line is a linear fit.(c) SPP dispersion relation for varying gra-phene Fermi levels as indicated. The squaresymbols indicate the range of experimentalλSPP from (b) reflecting the local variationsin dopant concentration.

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near-field excitations at higher photon energies and canreadily be studied using electron-based techniques [41,42].In conclusion, the full amplitude and phase spatially

resolved imaging of graphene SPPs that are scattered andreflected by discontinuities such as edges, grain boundaries,and defects provides nanoscale insight into local electronicstructure and dynamics in graphene. As the amplitude andphase couple differently to the reflection coefficients anddoping, their simultaneous measurement allows us todevelop a self-consistent phenomenological cavity modelwith no adjustable parameter to describe the SPP response.We observe the effects of charge pooling and associatedFermi level offsets in sample regions with a high densityof grain boundaries and folds. These results highlight theneed for nanoscale control of electronic structure for optimalgraphene optical and electronic device performance.

The authors thankGregAndreev for stimulating this study,as well as Joanna Atkin for fruitful discussions. This researchwas supported by the U.S. Department of Energy, Officeof Basic Energy Sciences, Division of Materials Sciencesand Engineering, under Award No. DE-FG02-12ER46893.J. A. G. and S. B. contributed equally to this work.

*Present address: National Institute of Standards and Tech-nology, Boulder, CO 80305, USA.

†markus.raschke@colorado.edu[1] E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418

(2007).[2] M. Jablan, H. Buljan, and M. Soljacic, Phys. Rev. B 80,

245435 (2009).[3] L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao,

H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang,Nat. Nanotechnol. 6, 630 (2011).

[4] H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag,W. Zhu, P. Avouris, and F. Xia, Nat. Nanotechnol. 7, 330(2012).

[5] R. Roldán and L. Brey, Phys. Rev. B 88, 115420 (2013).[6] A. N. Grigorenko, M. Polini, and K. S. Novoselov, Nat.

Photonics 6, 749 (2012).[7] K. A.VelizhaninandA.Efimov,Phys.Rev.B84,085401(2011).[8] F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo,

Nano Lett. 11, 3370 (2011).[9] B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D.W.

Pohl, Phys. Rev. Lett. 77, 1889 (1996).[10] X. Ren, A. Liu, C. Zou, L. Wang, Y. Cai, F. Sun, G. Guo,

and G. Guo, Appl. Phys. Lett. 98, 201113 (2011).[11] Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod,

C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M.Thiemens et al., Nano Lett. 11, 4701 (2011).

[12] Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod,M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G.Dominguez et al., Nature (London) 487, 82 (2012).

[13] J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri,F. Huth, J. Osmond,M. Spasenovic, A. Centeno, A. Pesquera,P. Godignon et al., Nature (London) 487, 77 (2012).

[14] Z. Fei, A. S. Rodin, W. Gannett, S. Dai, W. Regan,M. Wagner, M. K. Liu, A. S. McLeod, G. Dominguez,M. Thiemens et al., Nat. Nanotechnol. 8, 821 (2013).

[15] J. Chen, M. L. Nesterov, A. Y. Nikitin, S. Thongrattanasiri,P. Alonso-González, T. M. Slipchenko, F. Speck, M. Ostler,T. Seyller, I. Crassee et al., Nano Lett. 13, 6210 (2013).

[16] Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, andZ. X. Shen, ACS Nano 2, 2301 (2008).

[17] R. Sensarma, E. H. Hwang, and S. Das Sarma, Phys. Rev. B82, 195428 (2010).

[18] K. F. Mak, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 104,176404 (2010).

[19] Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin,P. Kim, H. L. Stormer, and D. N. Basov, Nat. Phys. 4, 532(2008).

[20] Handbook of Optical Constants of Solids, edited by E. Palik(Springer, Berlin, 2000).

[21] M. Jablan, M. Soljacic, and H. Buljan, Phys. Rev. B 83,161409(R) (2011).

[22] A. Politano, V. Formoso, and G. Chiarello, J. Phys.Condens. Matter 25, 345303 (2013).

[23] H. Yan, T. Low,W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea,P. Avouris, and F. Xia, Nat. Photonics 7, 394 (2013).

[24] V.W. Brar, M. S. Jang, M. Scherrott, J. J. Lopez, andH. A. Atwater, Nano Lett. 13, 2541 (2013).

[25] F. Keilmann and R. Hillenbrand, Phil. Trans. R. Soc. A 362,787 (2004).

[26] S.Aubert, A. Bruyant, S. Blaize, R. Bachelot, G. Lerondel, S.Hudlet, and P. Royer, J. Opt. Soc. Am. B 20, 2117 (2003).

[27] A. C. Jones, R. L. Olmon, S. E. Skrabalak, B. J. Wiley,Y. N. Xia, and M. B. Raschke, Nano Lett. 9, 2553 (2009).

[28] S. Berweger, D. M. Nguyen, E. A. Muller, H. A. Bechtel,T. T. Perkins, and M. B. Raschke, J. Am. Chem. Soc. 135,18292 (2013).

[29] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V.Khotevich, S. V. Morozov, and A. K. Geim, Proc. Natl.Acad. Sci. U.S.A. 102, 10451 (2005).

[30] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.113.055502 for furtherdiscussion of the model and observed phenomena, and dataof SPP scattering from a point-like defect.

[31] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, andA. K.Geim,Rev.Mod. Phys. 81, 109 (2009).

[32] L.Tapaszto,P.Nemes-Incze,G.Dobrik,K. J.Yoo,C.Hwang,and L. O. Biro, Appl. Phys. Lett. 100, 053114 (2012).

[33] A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 (2007).[34] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet,

K. von Klitzing, and A. Yacoby, Nat. Phys. 4, 144 (2008).[35] Y. Zhang, V.W. Brar, C. Girit, A. Zettl, and M. F. Crommie,

Nat. Phys. 5, 722 (2009).[36] A. Principi, G. Vignale, M. Carrega, and M. Polini, Phys.

Rev. B 88, 121405(R) (2013).[37] J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L.

Koppens, and F. J. García de Abajo, ACS Nano 6, 431(2012).

[38] Y. Gao and Z. Yuan, Solid State Commun. 151, 1009 (2011).[39] M. Pisarra, A. Sindona, P. Riccardi, V. M. Silkin, and

J. M. Pitarke, arXiv:1306.6273v3.[40] A. Politano and G. Chiarello, Carbon 71, 176 (2014).[41] T. Langer, D. F. Förster, C. Busse, T. Michely, H. Pfnür, and

C. Tegenkamp, New J. Phys. 13, 053006 (2011).[42] C. Tegenkamp, H. Pfnür, T. Langer, J. Baringhaus, and

H.W. Schumacher, J. Phys. Condens. Matter 23, 012001(2011).

PRL 113, 055502 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending

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055502-5

http://dx.doi.org/10.1103/PhysRevB.75.205418http://dx.doi.org/10.1103/PhysRevB.75.205418http://dx.doi.org/10.1103/PhysRevB.80.245435http://dx.doi.org/10.1103/PhysRevB.80.245435http://dx.doi.org/10.1038/nnano.2011.146http://dx.doi.org/10.1038/nnano.2012.59http://dx.doi.org/10.1038/nnano.2012.59http://dx.doi.org/10.1103/PhysRevB.88.115420http://dx.doi.org/10.1038/nphoton.2012.262http://dx.doi.org/10.1038/nphoton.2012.262http://dx.doi.org/10.1103/PhysRevB.84.085401http://dx.doi.org/10.1021/nl201771hhttp://dx.doi.org/10.1103/PhysRevLett.77.1889http://dx.doi.org/10.1063/1.3592757http://dx.doi.org/10.1021/nl202362dhttp://dx.doi.org/10.1038/nature11253http://dx.doi.org/10.1038/nature11254http://dx.doi.org/10.1038/nnano.2013.197http://dx.doi.org/10.1021/nl403622thttp://dx.doi.org/10.1021/nn800459ehttp://dx.doi.org/10.1103/PhysRevB.82.195428http://dx.doi.org/10.1103/PhysRevB.82.195428http://dx.doi.org/10.1103/PhysRevLett.104.176404http://dx.doi.org/10.1103/PhysRevLett.104.176404http://dx.doi.org/10.1038/nphys989http://dx.doi.org/10.1038/nphys989http://dx.doi.org/10.1103/PhysRevB.83.161409http://dx.doi.org/10.1103/PhysRevB.83.161409http://dx.doi.org/10.1088/0953-8984/25/34/345303http://dx.doi.org/10.1088/0953-8984/25/34/345303http://dx.doi.org/10.1038/nphoton.2013.57http://dx.doi.org/10.1021/nl400601chttp://dx.doi.org/10.1098/rsta.2003.1347http://dx.doi.org/10.1098/rsta.2003.1347http://dx.doi.org/10.1364/JOSAB.20.002117http://dx.doi.org/10.1021/nl900638phttp://dx.doi.org/10.1021/ja409815ghttp://dx.doi.org/10.1021/ja409815ghttp://dx.doi.org/10.1073/pnas.0502848102http://dx.doi.org/10.1073/pnas.0502848102http://link.aps.org/supplemental/10.1103/PhysRevLett.113.055502http://link.aps.org/supplemental/10.1103/PhysRevLett.113.055502http://link.aps.org/supplemental/10.1103/PhysRevLett.113.055502http://link.aps.org/supplemental/10.1103/PhysRevLett.113.055502http://link.aps.org/supplemental/10.1103/PhysRevLett.113.055502http://link.aps.org/supplemental/10.1103/PhysRevLett.113.055502http://link.aps.org/supplemental/10.1103/PhysRevLett.113.055502http://dx.doi.org/10.1103/RevModPhys.81.109http://dx.doi.org/10.1063/1.3681375http://dx.doi.org/10.1038/nmat1849http://dx.doi.org/10.1038/nphys781http://dx.doi.org/10.1038/nphys1365http://dx.doi.org/10.1103/PhysRevB.88.121405http://dx.doi.org/10.1103/PhysRevB.88.121405http://dx.doi.org/10.1021/nn2037626http://dx.doi.org/10.1021/nn2037626http://dx.doi.org/10.1016/j.ssc.2011.05.001http://arXiv.org/abs/1306.6273v3http://dx.doi.org/10.1016/j.carbon.2014.01.026http://dx.doi.org/10.1088/1367-2630/13/5/053006http://dx.doi.org/10.1088/0953-8984/23/1/012001http://dx.doi.org/10.1088/0953-8984/23/1/012001